Average Error: 28.0 → 2.5
Time: 42.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}} \cdot \frac{1}{\left(cos \cdot sin\right) \cdot x}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}} \cdot \frac{1}{\left(cos \cdot sin\right) \cdot x}
double f(double x, double cos, double sin) {
        double r1352968 = 2.0;
        double r1352969 = x;
        double r1352970 = r1352968 * r1352969;
        double r1352971 = cos(r1352970);
        double r1352972 = cos;
        double r1352973 = pow(r1352972, r1352968);
        double r1352974 = sin;
        double r1352975 = pow(r1352974, r1352968);
        double r1352976 = r1352969 * r1352975;
        double r1352977 = r1352976 * r1352969;
        double r1352978 = r1352973 * r1352977;
        double r1352979 = r1352971 / r1352978;
        return r1352979;
}
↓
double f(double x, double cos, double sin) {
        double r1352980 = 1.0;
        double r1352981 = cos;
        double r1352982 = sin;
        double r1352983 = r1352981 * r1352982;
        double r1352984 = x;
        double r1352985 = r1352983 * r1352984;
        double r1352986 = 2.0;
        double r1352987 = r1352986 * r1352984;
        double r1352988 = cos(r1352987);
        double r1352989 = r1352985 / r1352988;
        double r1352990 = r1352980 / r1352989;
        double r1352991 = r1352980 / r1352985;
        double r1352992 = r1352990 * r1352991;
        return r1352992;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.0

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

  2. Simplified2.8

    \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(\left(sin \cdot cos\right) \cdot x\right) \cdot \left(\left(sin \cdot cos\right) \cdot x\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity2.8

    \[\leadsto \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{\left(\left(sin \cdot cos\right) \cdot x\right) \cdot \left(\left(sin \cdot cos\right) \cdot x\right)}\]

  5. Applied times-frac2.5

    \[\leadsto \color{blue}{\frac{1}{\left(sin \cdot cos\right) \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot cos\right) \cdot x}}\]
  6. Using strategy rm

  7. Applied clear-num2.5

    \[\leadsto \frac{1}{\left(sin \cdot cos\right) \cdot x} \cdot \color{blue}{\frac{1}{\frac{\left(sin \cdot cos\right) \cdot x}{\cos \left(2 \cdot x\right)}}}\]

  8. Final simplification2.5

    \[\leadsto \frac{1}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}} \cdot \frac{1}{\left(cos \cdot sin\right) \cdot x}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))